Primitivity, Uniform Minimality, and State Complexity of Boolean Operations

Article
  • 6 Downloads

Abstract

A minimal deterministic finite automaton (DFA) is uniformly minimal if it always remains minimal when the final state set is replaced by a non-empty proper subset of the state set. We prove that a permutation DFA is uniformly minimal if and only if its transition monoid is a primitive group. We use this to study Boolean operations on group languages, which are recognized by direct products of permutation DFAs. A direct product cannot be uniformly minimal, except in the trivial case where one of the DFAs in the product is a one-state DFA. However, non-trivial direct products can satisfy a weaker condition we call uniform Boolean minimality, where only final state sets used to recognize Boolean operations are considered. We give sufficient conditions for a direct product of two DFAs to be uniformly Boolean minimal, which in turn gives sufficient conditions for pairs of group languages to have maximal state complexity under all binary Boolean operations (“maximal Boolean complexity”). In the case of permutation DFAs with one final state, we give necessary and sufficient conditions for pairs of group languages to have maximal Boolean complexity. Our results demonstrate a connection between primitive groups and automata with strong minimality properties.

Keywords

State complexity Finite automaton Primitive group Uniformly minimal Boolean operation Transition monoid 

Notes

Acknowledgements

I thank Jason Bell, Janusz Brzozowski and an anonymous referee for careful proofreading and helpful comments. The computer algebra system GAP [19] was invaluable for this research; I cannot overstate its importance in obtaining these results. In particular, I thank the authors of the Automata GAP package [13] and all contributors to GAP’s library of primitive groups.

References

  1. 1.
    Almeida, J., Rodaro, E.: Semisimple synchronizing automata and the Wedderburn-Artin theory. Int. J. Found. Comput. Sc. 27(02), 127–145 (2016).  https://doi.org/10.1142/S0129054116400037 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Araújo, J., Cameron, P.J., Steinberg, B.: Between primitive and 2-transitive: synchronization and its friends. arXiv:1511.03184 (2015)
  3. 3.
    Bell, J., Brzozowski, J., Moreira, N., Reis, R.: Symmetric groups and quotient complexity of boolean operations. In: Esparza, J., et al. (eds.) ICALP 2014, LNCS, vol. 8573, pp 1–12. Springer (2014).  https://doi.org/10.1007/978-3-662-43951-7_1
  4. 4.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symbolic Comput. 24(3–4), 235–265 (1997).  https://doi.org/10.1006/jsco.1996.0125 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brzozowski, J.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010).  https://doi.org/10.4204/EPTCS.3.2 MATHGoogle Scholar
  6. 6.
    Brzozowski, J.: In search of most complex regular languages. Int. J. Found. Comput. Sci. 24(06), 691–708 (2013).  https://doi.org/10.1142/S0129054113400133 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brzozowski, J.: Unrestricted state complexity of binary operations on regular languages. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016, LNCS, vol. 9777, pp 60–72. Springer (2016).  https://doi.org/10.1007/978-3-319-41114-9_5
  8. 8.
    Brzozowski, J., Jirásková, G., Li, B.: Quotient complexity of ideal languages. Theoret. Comput. Sci. 470, 36–52 (2013).  https://doi.org/10.1016/j.tcs.2012.10.055 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Brzozowski, J., Liu, B.: Quotient complexity of star-free languages. Int. J. Found. Comput. Sci. 23(06), 1261–1276 (2012).  https://doi.org/10.1142/S0129054112400515 MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Brzozowski, J.A., Davies, S., Liu, B.Y.V.: Most complex regular ideal languages. Discret. Math. Theoret. Comput. Sci. 18(3). Paper #15. https://dmtcs.episciences.org/volume/view/id/184 (2016)
  11. 11.
    Brzozowski, J.A., Sinnamon, C.: Unrestricted state complexity of binary operations on regular and ideal languages. J. Autom. Lang. Comb. 22(1–3), 29–59 (2017).  https://doi.org/10.25596/jalc-2017-029 MathSciNetGoogle Scholar
  12. 12.
    Cameron, P.J.: Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13(1), 1–22 (1981).  https://doi.org/10.1112/blms/13.1.1 MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Delgado, M., Linton, S., Morais, J.: Automata – a GAP package, Version 1.13. http://cmup.fc.up.pt/cmup/mdelgado/automata/ (2011)
  14. 14.
    Dixon, J.D., Mortimer, B.: Permutation groups. Springer, Berlin (1996).  https://doi.org/10.1007/978-1-4612-0731-3  https://doi.org/10.1007/978-1-4612-0731-3 CrossRefMATHGoogle Scholar
  15. 15.
    Dummit, D., Foote, R.: Abstract Algebra. Wiley, NY (2004)MATHGoogle Scholar
  16. 16.
    Ésik, Z., Gao, Y., Liu, G., Yu, S.: Estimation of state complexity of combined operations. Theoret. Comput. Sci. 410(35), 3272–3280 (2009).  https://doi.org/10.1016/j.tcs.2009.03.026 MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gao, Y., Moreira, N., Reis, R., Yu, S.: A survey on operational state complexity. J. Autom. Lang. Comb. 21(4), 251–310 (2016).  https://doi.org/10.25596/jalc-2016-251 MathSciNetMATHGoogle Scholar
  18. 18.
    Gao, Y., Yu, S.: State complexity and approximation. Int. J. Found. Comput. Sci. 23(05), 1085–1098 (2012).  https://doi.org/10.1142/S0129054112400461 MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    The GAP Group: GAP – Groups, Algorithms, and Programming, Version 4.8.6 (2016). http://www.gap-system.org
  20. 20.
    Kiraz, G.A.: Compressed storage of sparse finite-state transducers. In: Boldt, O., Jürgensen, H. (eds.) WIA 1999, LNCS, vol. 2214, pp 109–121. Springer (2001).  https://doi.org/10.1007/3-540-45526-4_11
  21. 21.
    Maslov, A.N.: Estimates of the number of states of finite automata. Dokl. Akad. Nauk SSSR 194, 1266–1268 (Russian) (1970). English translation: Soviet Math. Dokl. 11 (1970) 1373–1375Google Scholar
  22. 22.
    Restivo, A., Vaglica, R.: Extremal minimality conditions on automata. Theor. Comput. Sci. 440–441, 73–84 (2012).  https://doi.org/10.1016/j.tcs.2012.03.049 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Restivo, A., Vaglica, R.: A graph theoretic approach to automata minimality. Theoret. Comput. Sci. 429, 282–291 (2012).  https://doi.org/10.1016/j.tcs.2011.12.049 MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Salomaa, A., Salomaa, K., Yu, S.: State complexity of combined operations. Theoret. Comput. Sci. 383(2), 140–152 (2007).  https://doi.org/10.1016/j.tcs.2007.04.015 MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320(2), 315–329 (2004).  https://doi.org/10.1016/j.tcs.2004.02.032 MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Steinberg, B.: A theory of transformation monoids: combinatorics and representation theory. arXiv:https://arxiv.org/abs/1004.2982 (2010)
  27. 27.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theor. Comput. Sci. 125(2), 315–328 (1994).  https://doi.org/10.1016/0304-3975(92)00011-F MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

Personalised recommendations